DV01 Approximation Formula

The risk of a swap can be redistributed to more liquid swap maturities. The assignment of a bucketís swap risk into two (or more) nodes is done on a linearly interpolated continuously compounded yield.

For example, if we consider the risk at a node of maturity t2 to be redistributed between the nodes t1 and t3, the associated DV01 in t2 (Δ2) can be expressed as a function of the maturities t1 and t3, the associated DV01s (Δ1 & Δ3) and the Par Swap rates.

The question is how to derive this appoximated formula.

Question) Show that Δ2 can be approximated using the following formula (in the example above r1 = 59% & r2 = 41%):

Note that: t1, t2 and t3 represent the maturities in years; Y is the Par Swap rate and Δ is the DV01. Furthermore, we know that the following relationships are true for the discount factor, df, for a given maturity t & the continuously compounded zero yield y.